Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Not every $ p$-group can be generated by elements of the same order

Author(s): E. A. O'Brien; Carlo M. Scoppola; M. R. Vaughan-Lee
Journal: Proc. Amer. Math. Soc. 134 (2006), 3457-3464.
MSC (2000): Primary 20D15
Posted: June 12, 2006
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: For every prime $ p$, we exhibit a finite $ p$-group which cannot be generated by a set of elements, all having the same order. This answers a long-standing question from the Kourovka Notebook.

References:

1.
C. Baginski, Some remarks on finite $ p$-groups. Demonstratio Math. 14 (1981), 279-285. MR 0632286 (83f:20017)

2.
A. Caranti and C.M. Scoppola, Endomorphisms of two-generated metabelian groups that induce the identity modulo the derived subgroup. Arch. Math. 56 (1991), 218-227. MR 1091874 (92b:20038)

3.
Group Pub Forum (http://www.bath.ac.uk/$ ^\sim$masgcs/gpf.html)+, 2003.

4.
Marshall Hall, Jr., The theory of groups. The Macmillan Co., New York, 1959. MR 0103215 (21:1996)

5.
The Kourovka Notebook. Unsolved problems in group theory. Fifteenth augmented edition, 2002. Edited by V.D. Mazurov and E.I. Khukhro. MR 1956290 (2004f:20001a)

6.
Alexander Lubotzky and Avinoam Mann, Powerful $ p$-groups. I. Finite groups. J. Algebra 105 (1987), 484-505. MR 0873681 (88f:20045)

7.
M.F. Newman and E.A. O'Brien, Application of computers to questions like those of Burnside, II. Internat. J. Algebra Comput. 6 (1996), 593-605. MR 1419133 (97k:20002)

8.
E.A. O'Brien, The p-group generation algorithm. J. Symbolic Comput. 9 (1990), 677-698. MR 1075431 (91j:20050)

9.
Charles C. Sims, Computation with finitely presented groups. Cambridge University Press, 1994. MR 1267733 (95f:20053)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20D15

Retrieve articles in all Journals with MSC (2000): 20D15

Additional Information:

E. A. O'Brien
Affiliation: Department of Mathematics, University of Auckland, Auckland, New Zealand
Email: obrien@math.auckland.ac.nz

Carlo M. Scoppola
Affiliation: Dipartimento di Matematica Pura ed Applicata, Universita di L'Aquila, Coppito 67010, L'Aquila, Italy
Email: scoppola@univaq.it

M. R. Vaughan-Lee
Affiliation: Christ Church, University of Oxford, OX1 1DP, United Kingdom
Email: michael.vaughan-lee@christ-church.oxford.ac.uk

DOI: 10.1090/S0002-9939-06-08576-5
PII: S 0002-9939(06)08576-5
Received by editor(s): July 6, 2005
Posted: June 12, 2006
Additional Notes: The first author was supported by GNSAGA-INdAM and by EPSRC grant GR/S86259/01 while this paper was written
The second author is a member of GNSAGA-INdAM
Communicated by: Jonathan I. Hall
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google